International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
Acyclic Coloring of Middle and Total graph of Extended Duplicate graph of Ladder Graph C. Shobana Sarma1, K. Thirusangu2 1Department of Mathematics, Bharathi Women’s College (Autonomous) Chennai – 600 108, India 2Department of Mathematics, S.I.V.E.T. College Chennai 600 073, India
Abstract—In this paper we present acyclic coloring algorithms to color the vertices of middle graph and total graph of extended Duplicate graph of ladder Lm. Also we obtain the chromatic numbers of the same.
Keywords—Acyclic coloring, chromatic number, middle graph, total graph, extended duplicate graph, ladder graph.
I. INTRODUCTION A proper coloring of a graph G is the coloring of the vertices of G such that no two neighbors in G are assigned the same color. Thoughout this paper, by a graph we mean a finite, undirected, simple graph and the term coloring is used to denote vertex coloring of graphs. A acyclic coloring of a graph G is the proper vertex coloring such that the subgraph induced by any two color classes does not contains a cycle. The notion of acyclic chromatic number was introduced by B.Grunbaum in 1973[4]. The acyclic chromatic number of a graph G = G(V, E) is the minimum number of colors which are necessary to color G acyclically and is denoted by a(G). The acycling coloring of middle and total graphs of some class of graphs have been studied in the literature [1,5,6,7,8]. A ladder graph Lm is a planar undirected graph with 2m vertices and 3m-2 edges. It is obtained as the cartesian product of two path graphs, one of which has only one edge : Lm=Pm X P1, where m is the number of rungs in the ladder. The concept of extended duplicate graph was introduced by Thirusangu, et al. [10]. A duplicate graph of G is
DG = (V1, E1) where the vertex set V1 = V V and V V= and f : V V is bijective (for v V, we write f(v) = v) and the edge set E1 of DG is defined as follows. The edge uv is in E if and only if both uv and uv are edges in E1. The extended duplicate graph of DG, denoted by EDG, is defined as, adding an edge between any vertex from V to any other vertex in V', except the terminal vertices of V and V'. For convenience, we take v2V and v'2V and thus the edge v2v'2 is formed. The middle graph of G, denoted by M(G) was introduced by Michalak[2], in 1981 and is defined as follows. The vertex set of M(G) is V(G) E(G). Any two vertices x, y in M(G) are adjacent in M(G) if one of the following case holds. (i) x and y are adjacent edges in G (ii) x and y are incident in G. The total graph of G, denoted by T(G) was introduced by Michalak(1981) and Harary(1969)[2,3] and is defined as follows. The vertex set of T(G) is V(G) E(G). Any two vertices x, y in the vertex set of T(G) are adjacent in T(G) if one of the following cases holds. (i) x and y are adjacent vertices in G. (ii) x and y are adjacent edges in G. (iii) x and y are incident in G.
A Duplicate graph of Lm DG(Lm) contains 4m vertices and 6m-4 edges. A duplicate graph of Ladder graph have been studied in the literature [9]. The extended duplicate graph ofLm, denoted by EDG(Lm) with 4m vertices and 6m-3 edges, is obtained from the duplicate graph of ladder by joining the vertices v2 and v2. This document is a template. An electronic copy can be downloaded from the conference website. For questions on paper guidelines, please contact the conference publications committee as indicated on the conference website. Information about final paper submission is available from the conference website.
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International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
II. ACYCLIC COLORING OF M[EDG(LM)] Coloring Algorithm 1: Input: M[EDG(Lm)], m 3 V {v1, v2, …, v2m, vꞌ1, vꞌ2, …, vꞌ2m ,x1,x2,....,x6m-3} { for (k = 1 to 2m) {
vk, vꞌk 1; } for (k =1 to m) {
x2k-1 2; x3(m-1)+2k 2; } for (k =1 to m-1) {
x2k 3; x3m+2(k-1) 3; } for (k = 1 to m ) 2 {
x2(m+k-1) 4; x5m+2(k-2) 4; } for (k = 1 to m 1 ) 2 {
x2(m+k)-1 5; x5m+2k-3 5; }
x6m-3 5; } Output: vertex colored M[EDG(Lm)].
Theorem 1:The acyclic chromatic number of middle graph of extended duplicate graph of ladder Lm is given by a(M[EDG(Lm)]) = 5, m 3. ProofColor the vertices of M[EDG(Lm)] as given in the algorithm 1. The color class of 1 is { vk, vꞌk ; 1 k 2m}. The color class of 2 and 3 are {x2k-1 , x3(m-1)+2k ; 1 k 2m} and
{ x2k , xm+2k+4 ; 1 k m-1} respectively. The color class of 4 and 5 are {x2(m+k-1) , x5m+2(k-2) ; 1 k } and
{ x2(m+k)-1 , x5m+2k-3 , x6m-3 ; 1 k } respectively.
Case (i) Consider the color classes of 1 and 2. The induced subgraph of the color classes of 1 and 2 is a collection of paths P3 and therefore, it is an acyclic graph. Case (ii) Consider the color classes of 1 and 3. The induced subgraph of the color classes of 1 and 3 is a collection of paths P3 and therefore, it is an acyclic graph. Case (iii)Consider the color classes of 1 and 4. The induced subgraph of the color classes of 1 and 4 is a collection of paths P3 and isolated vertices, therefore, it is an acyclic graph. Case (iv)Consider the color classes of 1 and 5. The induced subgraph of the color classes of 1 and 5 is a collection of paths P3 and isolated vertices, therefore, it is an acyclic graph. Case (v)Consider the color classes of 2 and 3. The induced subgraph of the color classes of 2 and 3 is a collection {x1 x2 x3 ...... x2m-1} and {x3n-1 x3n x3n+1 ...... x5m-3} of paths and therefore, it is an acyclic graph. Case (vi)Consider the color classes of 2 and 4. The induced subgraph of the color classes of 2 and 4 is a collection of paths P3, P5 and isolated vertices, therefore, it is an acyclic graph. Case (vii)Consider the color classes of 2 and 5. The induced subgraph of the color classes of 2 and 5 is a collection of paths P3, P5 and isolated vertices, therefore, it is an acyclic graph.
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International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
Case (viii)Consider the color classes of 3 and 4. The induced subgraph of the color classes of 3 and 4 is a collection of paths P3 and isolated vertices, therefore, it is an acyclic graph.
Case (ix)Consider the color classes of 3 and 5. The induced subgraph of the color classes of 3 and 5 is a collection of paths P3 and isolated vertices, therefore, it is an acyclic graph. Case (x)Consider the color classes of 4 and 5. The induced subgraph of the color classes of 4 and 5 is a collection of paths {x2m x2m+1} and {x6m-3 x5m-2}therefore, it is an acyclic graph. Thus, the induced subgraph of any two color classes is acyclic and therefore the coloring given in the algorithm is an acyclic coloring. Hence a(M[EDG(Lm)]) = 5, m 3.
III. ACYCLIC COLORING OF T[EDG(LM)] Coloring Algorithm 2: Input: M[EDG(Lm)], m 4 V {v1, v2, …, v2m, vꞌ1, vꞌ2, …, vꞌ2m ,x1,x2,....,x6m-3} { for (k = 1 to m ) 2 { if m 1 (mod 2)
v1, v4k, v4k+1 1; else
v1, v4k 1; for (r = 0 to m -1) 2
xm+2r+1 1; } for (k = -1 to m ) 3 { if m 1 (mod 2) for (r = 0 to )
{
x3m+2k+1 , v3+4r 1; } else for (r = 0 to -1)
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International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
{
x3m+2k+1 , vꞌ5+4r 1;
} } if m 1 (mod 2) for (k =0 to m ) 2 { for (r = 1 to )
{
vꞌ2+4k, xm+2r 2; } } for (k = 0 to m-1) {
x3m+2k 2; } else for (k =0 to m ) 3 { for (r = 1 to m ) 3 {
v3, x2m, vꞌ4k+2, vꞌ4r+3 2; } for (k = 0 to m-2) {
x3m+2k 2; } } for (k = 1 to m +1) 2 { if m 1 (mod 2) for (r = 0 to m ) 3 {
x2k-1 , vꞌ4r+3 3; for (r = 1 to m ) 2
vꞌ1, vꞌ4r, vꞌ4r+1 3; } else for (r = 0 to m -1) 3 {
x2k-1 , v4r+5 3; } for (k = 1 to )
{
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International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
for (r = 1 to (m 2) ) 2
vꞌ1, vꞌ4k, x4m+2r -1 3; } } if m 1 (mod 2) for (k =0 to m ) 2 { for (r = 0 to -1)
v4k+2, x4m+2r 4; } for (k = 1 to m) {
x2k 4; } else for (k = 1 to m -1) 2 {
v2, x5m-2, v4k+2, v4k+3 4; } for (k = 1 to m-1) {
x2k, vꞌ3 4; } } if m 1 (mod 2) for (k = 1 to )
{ for (r = 0 to -1)
x2m+2k-1, x5m+2r-1 5; } else for (k =1 to -1)
{ for (r = 0 to -2)
{
x2m+2k-1, x5m+2r-1 5; } for (k = 1 to -1)
{ for (r = 0 to -2)
{
x2m+2k, x5m+2r 6; } x6m-3 6; }
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International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
Output: vertex colored T[EDG(Lm)].
Theorem 2: The acyclic chromatic number of total graph of extended duplicate graph of ladder Lm is given by a(T[EDG(Lm)]) = 6, m 4. Proof Color the vertices of T[EDG(Lm)] as given in the algorithm 2. Case (i) Consider the color classes of 1 and 2. The induced subgraph of the color classes of 1 and 2 is a collection of paths and trees, therefore, it is an acyclic graph. Case (ii) Consider the color classes of 1 and 3. The induced subgraph of the color classes of 1 and 3 is a collection of trees and stars, therefore, it is an acyclic graph. Case (iii) Consider the color classes of 1 and 4. The induced subgraph of the color classes of 1 and 4 is a collection of stars when m is odd and collection of stars and paths when m is even and therefore, it is an acyclic graph. Case (iv) Consider the color classes of 1 and 5. The induced subgraph of the color classes of 1 and 5 is a collection of paths and isolated vertices, therefore, it is an acyclic graph. Case (v) Consider the color classes of 1 and 6. The induced subgraph of the color classes of 1 and 6 is a collection of paths and isolated vertices, therefore, it is an acyclic graph. Case (vi) Consider the color classes of 2 and 3. The induced subgraph of the color classes of 2 and 3 is a collection of paths when m is odd and collection of paths and trees when m is even and therefore, it is an acyclic graph. Case (vii) Consider the color classes of 2 and 4. The induced subgraph of the color classes of 2 and 4 is a collection of stars and paths, therefore, it is an acyclic graph. Case (viii) Consider the color classes of 2 and 5. The induced subgraph of the color classes of 2 and 5 is a collection of paths, therefore, it is an acyclic graph. Case (ix) Consider the color classes of 2 and 6. The induced subgraph of the color classes of 3 and 4 is a collection of paths, therefore, it is an acyclic graph.
Case (x) Consider the color classes of 3 and 4. The induced subgraph of the color classes of 3 and 4 is a collection of paths, therefore, it is an acyclic graph. Case (xi) Consider the color classes of 3 and 5. The induced subgraph of the color classes of 3 and 5 is a collection of paths, therefore, it is an acyclic graph. Case (xii) Consider the color classes of 3 and 6. The induced subgraph of the color classes of 3 and 6 is a collection of paths, therefore, it is an acyclic graph. Case (xiii) Consider the color classes of 4 and 5. The induced subgraph of the color classes of 4 and 5 is a collection of paths, therefore, it is an acyclic graph. Case (xiv) Consider the color classes of 4 and 6. The induced subgraph of the color classes of 4 and 6 is a collection of paths, therefore, it is an acyclic graph. Case (xv) Consider the color classes of 5 and 6. The induced subgraph of the color classes of 5 and 6 is a null graph, therefore, it is an acyclic graph.
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International Journal of Mathematics Trends and Technology (IJMTT) – Special Issue NCCFQET May 2018
Thus, the induced subgraph of any two color classes is acyclic and therefore the coloring given in the algorithm is an acyclic coloring. Hence a(T[EDG(Lm)]) = 6, m 4.
IV. CONCLUSIONS In this paper, we obtained the acyclic chromatic number of of middle graph and total graph of extended Duplicate graph of ladder Lm.
REFERENCES [1] R. Arundhadhi, and R. Sattanathan, “Acyclic and star coloring of bistar graph families,” International Journal of Scientific and Research Publications (IJSRP), vol. 2(3), pp. 14, 2012. [2] Danuta Michalak, “On middle and total graphs with coarseness number equal 1,” Spinger Verlag Graph Theory, Lagow proceedings, Berlin Heidelberg, New York, Tokyo, vol. 1018, pp. 139150, 1981. [3] Frank Harary, Graph Theory, Narosa Publishing Home, 2001. [4] B. Grunbaum, “Acyclic coloring of planar graphs,” Israel J. Math., vol. 14(3), pp. 390408, 1973. [5] C. Shobana Sarma, and K. Thirusangu, “Acyclic coloring of extended duplicate graph of path graph families,” International Journal of Applied Engineering Research, ISSN 09734562, vol. 10(72), 2015. [6] C. Shobana Sarma, and K. Thirusangu, “Acyclic coloring of Extended duplicate graph of Cycle graph families,”International Journal of Pure and Applied Mathematics,ISSN 1311-8080, vol. 113(9), pp. 192200, 2017. [7] C. Shobana Sarma, and K. Thirusangu, “Acyclic and Star coloring of duplicate graph of Ladder graph,”Proceedings of National Conference on Recent Trends in Hybridization of Mathematical Science (RETHMS – 2017). [8] K. Thilagavathi, and Shahnas Banu, “Acyclic coloring of star graph families,” International Journal of Computer Applications, vol. 7(2), pp. 3133, 2010. [9] K. Thirusangu, P.P. Ulaganathan, and P. Vijaya kumar, “Some Cordial labeling of duplicate graph of Ladder graph,” Annals of Pure and Applied Mathematics, ISSN 2279-087X(P), vol. 8(2), pp. 4350, 2014. [10] P.P. Ulaganathan, K. Thirusangu, and B. Selvam, “Edge magic total labeling in extended duplicate graph of path,”International Journal of Applied Engineering Research,vol. 6(10), pp. 12111217, 2011.
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